Sec 2 · G2 (N(A))

Quadrilaterals and polygons

Practise quadrilaterals and polygons with step-by-step help. Free account.

1. Quadrilaterals

Quadrilateral	Sides	Angles	Diagonals
parallelogram	opposite sides parallel and equal	opposite angles equal	bisect each other
rectangle	opposite sides parallel and equal	all $90^\circ$	equal, bisect each other
rhombus	all equal	opposite angles equal	bisect each other at $90^\circ$
square	all equal	all $90^\circ$	equal, bisect each other at $90^\circ$
trapezium	one pair parallel
isosceles trapezium	one pair parallel; the other two sides equal	base angles equal	equal
kite	two pairs of equal adjacent sides	one pair equal	cross at $90^\circ$ ; one bisects the other

📖 Worked example

$ABCD$ is a parallelogram. Find $x$ .

Step 1 of 2

Which angle fact gives x?

x^\circ + 113^\circ = 180^\circ

(int. ∠s,

AB \parallel DC

)

Angles in a ratio	Share	Example
$\angle A : \angle B : \angle C : \angle D$	$360^\circ$	$3 : 4 : 5 : 6$ : $18$ units $\rightarrow 360^\circ$ , $1$ unit $\rightarrow 20^\circ$
parallelogram, $\angle A : \angle B$	$180^\circ$ (int. $\angle$ s)	$2 : 3$ : $1$ unit $\rightarrow 36^\circ$

Practise this →

2. Polygons

$n$ sides	Formula	Example: hexagon, $n = 6$
sum of interior angles	$(n - 2) \times 180^\circ$	$720^\circ$
sum of exterior angles	$360^\circ$	$360^\circ$
each exterior angle (regular)	$\dfrac{360^\circ}{n}$	$60^\circ$
each interior angle (regular)	$180^\circ - \text{exterior angle}$	$120^\circ$

Number of sides of a regular polygon with interior angle 156°

\displaystyle \text{exterior angle} = 180^\circ - 156^\circ

e.g.

\displaystyle n = \dfrac{360}{180 - 156}

\displaystyle n = 15

📖 Worked example

The diagram shows a regular nonagon. Find $x$ .

Step 1 of 2

Sum of the interior angles of a nonagon

(9 - 2) \times 180^\circ = 1260^\circ

📖 Worked example

Each interior angle of a regular polygon is $3$ times each of its exterior angles. Find the number of sides of the polygon.

Step 1 of 3

An interior angle and its exterior angle make a straight line: together 180°

3 + 1 = 4

units

\rightarrow 180^\circ

1

unit

\rightarrow \dfrac{180}{4} = 45^\circ

📖 Worked example

In the diagram, a square, a regular pentagon and a regular polygon with $n$ sides fit together round the point $O$ , with no gaps. Find $n$ .

Step 1 of 3

Each interior angle of the two known polygons: (n − 2) × 180° ÷ n

Square:

90^\circ

Regular pentagon:

108^\circ

📖 Worked example

A polygon has $n$ sides. 3 of its interior angles are $100^\circ$ , $120^\circ$ and $140^\circ$ . Each of the other interior angles is $165^\circ$ . Find $n$ .

Step 1 of 3

Work with exterior angles (180° − each interior angle): they add up to 360°

Exterior angles of the given ones:

180^\circ - 100^\circ = 80^\circ

180^\circ - 120^\circ = 60^\circ

180^\circ - 140^\circ = 40^\circ

Their sum

= 180^\circ

📖 Worked example

The diagram shows a regular hexagon $ABCD\ldots$ with the diagonal $AC$ . Find $x$ .

Step 1 of 2

Each interior angle of the regular polygon

\angle ABC = \dfrac{(6 - 2) \times 180}{6}

\phantom{\angle ABC} = 120^\circ

Practise this →

3. Special quadrilaterals

The seven special quadrilaterals with their marks and dashed diagonals: a square (four equal sides, four right angles, diagonals at right angles); a rectangle (opposite sides equal, four right angles); a rhombus (four equal sides, diagonals at right angles); a parallelogram (both pairs of opposite sides parallel); a kite (two pairs of equal sides next to each other, diagonals at right angles); a trapezium (one pair of parallel sides); an isosceles trapezium (one pair of parallel sides, the other two sides equal, base angles both 64°).

Property	square	rectangle	rhombus	parallelogram	kite	trapezium	isosceles trapezium
all sides equal	✓	✗	✓	✗	✗	✗	✗
both pairs of opposite sides parallel	✓	✓	✓	✓	✗	✗	✗
only one pair of parallel sides	✗	✗	✗	✗	✗	✓	✓
all angles $90^\circ$	✓	✓	✗	✗	✗	✗	✗
opposite angles equal	✓	✓	✓	✓	✗	✗	✗
diagonals bisect each other	✓	✓	✓	✓	✗	✗	✗
diagonals equal	✓	✓	✗	✗	✗	✗	✓
diagonals cross at $90^\circ$	✓	✗	✓	✗	✓	✗	✗
diagonals bisect the angles	✓	✗	✓	✗	✗	✗	✗

📖 Worked example

The marks on the diagram show what is known about quadrilateral $ABCD$ . Write down the most special name of $ABCD$ .

Step 1 of 4

Equal sides (the ticks)

Equal sides: all four.

📖 Worked example

$ABCD$ is a rhombus. Its diagonals meet at $O$ . $AC = 12$ cm and $BD = 16$ cm. Find $AB$ .

Step 1 of 2

The diagonals of a rhombus bisect each other at right angles

OA = 6

cm,

OB = 8

\angle AOB = 90^\circ

📖 Worked example

$ABCD$ is a rectangle. Its diagonals meet at $O$ , and $\angle OAB = 28^\circ$ . Find $x$ .

Step 1 of 2

Find ∠OBA and give the reason

\angle OBA = 28^\circ

(base ∠s of isos. △)

The angles show	Sides not all equal	Four equal sides
all four angles $90^\circ$	rectangle	square
both pairs of opposite angles equal	parallelogram	rhombus
only $\angle A + \angle D = 180^\circ$ ( $AB \parallel DC$ )	trapezium

Practise this →

4. Line symmetry

Lines of symmetry dashed: a square has 4, a rectangle 2, an equilateral triangle 3, a rhombus 2 (its diagonals), a parallelogram 0, a kite 1, an isosceles trapezium 1 and a regular hexagon 6.

Word	Meaning
line of symmetry	a fold line: the two halves match exactly
regular polygon, $n$ sides	$n$ lines of symmetry

📖 Worked example

The diagram shows a kite. State whether the dashed line is a line of symmetry of the shape.

Step 1 of 2

Imagine folding the shape along the dashed line

Each corner on one side must land exactly on a corner on the other side, and each side on a side.

📖 Worked example

The diagram shows a rectangle. Write down the number of lines of symmetry of the shape.

Step 1 of 2

Look for every fold line that makes the two halves match

Lines of symmetry: the two lines joining the middles of opposite sides (not the diagonals).

Practise this →

5. Rotational symmetry

Each shape turned about its centre until a marked corner lands on the next corner: an equilateral triangle after 120° (order 3), a square after 90° (order 4), a parallelogram after 180° (order 2); a kite fits only after a full turn (order 1).

Order: how many times it fits in one full turn

\displaystyle \text{angle of rotation} = \dfrac{360^\circ}{\text{order}}

e.g.

\displaystyle \dfrac{360^\circ}{4}

\displaystyle = 90^\circ

a square: order $4$

Shape	Lines of symmetry	Order	Angle of rotation
square	$4$	$4$	$90^\circ$
rectangle	$2$	$2$	$180^\circ$
equilateral triangle	$3$	$3$	$120^\circ$
rhombus	$2$	$2$	$180^\circ$
parallelogram	$0$	$2$	$180^\circ$
kite	$1$	$1$
regular hexagon	$6$	$6$	$60^\circ$

📖 Worked example

The diagram shows a propeller shape. Write down its order of rotational symmetry.

Step 1 of 1

Turn the shape about its centre: in one full turn, how many times does it look exactly the same?

It looks the same every third of a turn.

\text{Order} = 3

📖 Worked example

The diagram shows an equilateral triangle. Find the smallest angle through which it can be rotated about its centre to look exactly the same.

Step 1 of 2

Turn the shape about its centre: in one full turn, how many times does it look exactly the same?

It looks the same every third of a turn.

\text{Order} = 3

Practise this →

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Sec 2 · G2 (N(A))

Quadrilaterals and polygons

Practise quadrilaterals and polygons with step-by-step help. Free account.

Start free

1. Quadrilaterals

Quadrilateral	Sides	Angles	Diagonals
parallelogram	opposite sides parallel and equal	opposite angles equal	bisect each other
rectangle	opposite sides parallel and equal	all $90^\circ$	equal, bisect each other
rhombus	all equal	opposite angles equal	bisect each other at $90^\circ$
square	all equal	all $90^\circ$	equal, bisect each other at $90^\circ$
trapezium	one pair parallel
isosceles trapezium	one pair parallel; the other two sides equal	base angles equal	equal
kite	two pairs of equal adjacent sides	one pair equal	cross at $90^\circ$ ; one bisects the other

📖 Worked example

$ABCD$ is a parallelogram. Find $x$ .

Step 1 of 2

Which angle fact gives x?

x^\circ + 113^\circ = 180^\circ

(int. ∠s,

AB \parallel DC

)

Angles in a ratio	Share	Example
$\angle A : \angle B : \angle C : \angle D$	$360^\circ$	$3 : 4 : 5 : 6$ : $18$ units $\rightarrow 360^\circ$ , $1$ unit $\rightarrow 20^\circ$
parallelogram, $\angle A : \angle B$	$180^\circ$ (int. $\angle$ s)	$2 : 3$ : $1$ unit $\rightarrow 36^\circ$

Practise this →

2. Polygons

$n$ sides	Formula	Example: hexagon, $n = 6$
sum of interior angles	$(n - 2) \times 180^\circ$	$720^\circ$
sum of exterior angles	$360^\circ$	$360^\circ$
each exterior angle (regular)	$\dfrac{360^\circ}{n}$	$60^\circ$
each interior angle (regular)	$180^\circ - \text{exterior angle}$	$120^\circ$

Number of sides of a regular polygon with interior angle 156°

\displaystyle \text{exterior angle} = 180^\circ - 156^\circ

e.g.

\displaystyle n = \dfrac{360}{180 - 156}

\displaystyle n = 15

📖 Worked example

The diagram shows a regular nonagon. Find $x$ .

Step 1 of 2

Sum of the interior angles of a nonagon

(9 - 2) \times 180^\circ = 1260^\circ

📖 Worked example

Each interior angle of a regular polygon is $3$ times each of its exterior angles. Find the number of sides of the polygon.

Step 1 of 3

An interior angle and its exterior angle make a straight line: together 180°

3 + 1 = 4

units

\rightarrow 180^\circ

1

unit

\rightarrow \dfrac{180}{4} = 45^\circ

📖 Worked example

In the diagram, a square, a regular pentagon and a regular polygon with $n$ sides fit together round the point $O$ , with no gaps. Find $n$ .

Step 1 of 3

Each interior angle of the two known polygons: (n − 2) × 180° ÷ n

Square:

90^\circ

Regular pentagon:

108^\circ

📖 Worked example

A polygon has $n$ sides. 3 of its interior angles are $100^\circ$ , $120^\circ$ and $140^\circ$ . Each of the other interior angles is $165^\circ$ . Find $n$ .

Step 1 of 3

Work with exterior angles (180° − each interior angle): they add up to 360°

Exterior angles of the given ones:

180^\circ - 100^\circ = 80^\circ

180^\circ - 120^\circ = 60^\circ

180^\circ - 140^\circ = 40^\circ

Their sum

= 180^\circ

📖 Worked example

The diagram shows a regular hexagon $ABCD\ldots$ with the diagonal $AC$ . Find $x$ .

Step 1 of 2

Each interior angle of the regular polygon

\angle ABC = \dfrac{(6 - 2) \times 180}{6}

\phantom{\angle ABC} = 120^\circ

Practise this →

3. Special quadrilaterals

Property	square	rectangle	rhombus	parallelogram	kite	trapezium	isosceles trapezium
all sides equal	✓	✗	✓	✗	✗	✗	✗
both pairs of opposite sides parallel	✓	✓	✓	✓	✗	✗	✗
only one pair of parallel sides	✗	✗	✗	✗	✗	✓	✓
all angles $90^\circ$	✓	✓	✗	✗	✗	✗	✗
opposite angles equal	✓	✓	✓	✓	✗	✗	✗
diagonals bisect each other	✓	✓	✓	✓	✗	✗	✗
diagonals equal	✓	✓	✗	✗	✗	✗	✓
diagonals cross at $90^\circ$	✓	✗	✓	✗	✓	✗	✗
diagonals bisect the angles	✓	✗	✓	✗	✗	✗	✗

📖 Worked example

The marks on the diagram show what is known about quadrilateral $ABCD$ . Write down the most special name of $ABCD$ .

Step 1 of 4

Equal sides (the ticks)

Equal sides: all four.

📖 Worked example

$ABCD$ is a rhombus. Its diagonals meet at $O$ . $AC = 12$ cm and $BD = 16$ cm. Find $AB$ .

Step 1 of 2

The diagonals of a rhombus bisect each other at right angles

OA = 6

cm,

OB = 8

\angle AOB = 90^\circ

📖 Worked example

$ABCD$ is a rectangle. Its diagonals meet at $O$ , and $\angle OAB = 28^\circ$ . Find $x$ .

Step 1 of 2

Find ∠OBA and give the reason

\angle OBA = 28^\circ

(base ∠s of isos. △)

The angles show	Sides not all equal	Four equal sides
all four angles $90^\circ$	rectangle	square
both pairs of opposite angles equal	parallelogram	rhombus
only $\angle A + \angle D = 180^\circ$ ( $AB \parallel DC$ )	trapezium

Practise this →

4. Line symmetry

Word	Meaning
line of symmetry	a fold line: the two halves match exactly
regular polygon, $n$ sides	$n$ lines of symmetry

📖 Worked example

The diagram shows a kite. State whether the dashed line is a line of symmetry of the shape.

Step 1 of 2

Imagine folding the shape along the dashed line

Each corner on one side must land exactly on a corner on the other side, and each side on a side.

📖 Worked example

The diagram shows a rectangle. Write down the number of lines of symmetry of the shape.

Step 1 of 2

Look for every fold line that makes the two halves match

Lines of symmetry: the two lines joining the middles of opposite sides (not the diagonals).

Practise this →

5. Rotational symmetry

Order: how many times it fits in one full turn

\displaystyle \text{angle of rotation} = \dfrac{360^\circ}{\text{order}}

e.g.

\displaystyle \dfrac{360^\circ}{4}

\displaystyle = 90^\circ

a square: order $4$

Shape	Lines of symmetry	Order	Angle of rotation
square	$4$	$4$	$90^\circ$
rectangle	$2$	$2$	$180^\circ$
equilateral triangle	$3$	$3$	$120^\circ$
rhombus	$2$	$2$	$180^\circ$
parallelogram	$0$	$2$	$180^\circ$
kite	$1$	$1$
regular hexagon	$6$	$6$	$60^\circ$

📖 Worked example

The diagram shows a propeller shape. Write down its order of rotational symmetry.

Step 1 of 1

Turn the shape about its centre: in one full turn, how many times does it look exactly the same?

It looks the same every third of a turn.

\text{Order} = 3

📖 Worked example

The diagram shows an equilateral triangle. Find the smallest angle through which it can be rotated about its centre to look exactly the same.

Step 1 of 2

Turn the shape about its centre: in one full turn, how many times does it look exactly the same?

It looks the same every third of a turn.

\text{Order} = 3

Practise this →

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